Three former presidents looked on at Washington National Cathedral as a fourth — George W Bush — eulogised his dad. "To us," the son said of the father, " he was the brightest of a thousand points of light."

Former US President George W Bush follows a military honor guard as they carry the casket out of a state funeral for US President George HW Bush at the Washington National Cathedral in Washington. Reuters

George W Bush broke down briefly at the end of his eulogy while invoking the daughter his parents lost when she was 3 and his mother, Barbara, who died in April. He took comfort in knowing "Dad is hugging Robin and holding Mom's hand again."

For all the sombre tributes to the late president's public service and strength of character, laughter filled the cathedral time after time. The late president's eulogists — son included — noted Bush's tendency to tangle his words and show his goofy side.

He was "the last great-soldier statesman," historian Jon Meacham said in his eulogy, "our shield" in dangerous times. But he also said that Bush, campaigning in a crowd in a department store, once shook hands with a mannequin. Rather than flushing in embarrassment, he simply cracked, "Never know. Gotta ask."

After the service, the hearse and a long procession of cars headed for Joint Base Andrews and the flight to Texas — but first down to the National Mall to pass by the World War II Memorial. The congregation at the cathedral, filled with foreign leaders and diplomats, Americans of high office and others touched by Bush's life, rose for the arrival of the casket, accompanied by clergy of faiths from around the world.

In their row together, President Donald Trump and former Presidents Barack Obama, Jimmy Carter and Bill Clinton stood with their spouses and all placed their hands over their hearts.

Meacham also praised Bush's call to volunteerism — his "1,000 points of light" — placing it alongside Abraham Lincoln's call to honour "the better angels of our nature" in the American rhetorical canon. Meacham called those lines "companion verses in America's national hymn."

Trump had mocked "1,000 points of light" last summer at a rally, saying "What the hell is that? Has anyone ever figured that one out? And it was put out by a Republican, wasn't it?"

Former Canadian prime minister Brian Mulroney praised Bush as a strong world leader who helped oversee the end of the Cold War and the collapse of the Soviet Union and set the stage for the North American Free Trade Agreement with Canada and Mexico, achieved under his successor, Clinton.

With Trump, a NAFTA critic, seated in the front row, Mulroney hailed the "largest and richest free trade area in the history of the world." The three countries have agreed on a revised trade agreement pushed by Trump.

Alan Simpson, a former Republican senator from Wyoming, regaled the congregation with stories from his years as Bush's friend in Washington. More seriously, he recalled that when he went through a rough patch in the political game, Bush conspicuously stood by him against the advice of aides. "You would have wanted him on your side," he said.

The national funeral service capped three days of remembrance in Washington before Bush's remains return to Texas on Wednesday for burial Thursday. He died Friday at age 94.

Bush will lie in repose at Saint Martin's Episcopal Church before burial at his family plot on the presidential library grounds at Texas A&M University in College Station. His final resting place will be alongside Barbara Bush, his wife of 73 years, and Robin Bush, the daughter they lost to leukaemia in 1953.

On Wednesday morning, a military band played "Hail to the Chief" as Bush's casket was carried down the steps of the US Capitol, where he had lain in state. Family members looked on as servicemen fired off a cannon salute. His hearse was then driven in a motorcade to the cathedral ceremony, slowing in front of the White House. Bush's route was lined with people much of the way, bundled in winter hats and taking photos.

Waiting for his arrival inside, Trump shook hands with Obama and former first lady Michelle Obama, who greeted him by saying "Good morning." Trump did not shake hands with Bill and Hillary Clinton, who looked straight ahead. Bill Clinton and Mrs Obama smiled and chatted as music played. Carter was seated silently next to Hillary Clinton in the cavernous cathedral. Obama cracked up laughing at someone's quip. Vice President Mike Pence shook Carter's hand.

Trump tweeted Wednesday that the day marked "a celebration for a great man who has led a long and distinguished life." Trump and his wife took their seats after the others, briefly greeting the Obamas seated next to them.

On Tuesday, soldiers, citizens in wheelchairs and long lines of others on foot wound through the Capitol Rotunda to view Bush's casket and honour a president whose legacy included World War military service and a landmark law affirming the rights of the disabled.

Former Senator Bob Dole, a compatriot in war, peace and political struggle, steadied himself out of his wheelchair and saluted his old friend and one-time rival.

Trump ordered the federal government closed Wednesday for a national day of mourning. Flags on public buildings are flying at half-staff for 30 days. As at notable moments in his life, Bush brought together Republicans and Democrats in his death, and not only the VIPs.